which places enjoy equal day and night throughout the year

Nowhere on Earth. But would be on the equator of any planet which has a circular orbit, and whose axis of rotation is perpendicular to the plane of the orbit. Days and nights are of equal length on the equator of any planet (including Earth). That's kind of what "equator" MEANS. In the situation described above, days and nights would be of equal length everywhere on the planet's surface (and even then the requirement that the orbit be circular is overkill).

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places near the equator ................

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